The algebraic degree of geometric optimization problems
Discrete & Computational Geometry
Algebraic optimization: the Fermat-Weber location problem
Mathematical Programming: Series A and B
Sublinear time algorithms for metric space problems
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Mobile facility location (extended abstract)
DIALM '00 Proceedings of the 4th international workshop on Discrete algorithms and methods for mobile computing and communications
Dynamic planar convex hull operations in near-logarithmic amortized time
Journal of the ACM (JACM)
Maintaining approximate extent measures of moving points
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Fast approximations for sums of distances, clustering and the Fermat--Weber problem
Computational Geometry: Theory and Applications
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
On the Continuous Fermat-Weber Problem
Operations Research
Smooth kinetic maintenance of clusters
Computational Geometry: Theory and Applications - Special issue on the 19th annual symposium on computational geometry - SoCG 2003
Geometric facility location under continuous motion: bounded-velocity approximations to the mobile euclidean k-centre and k-median problems
Solving the robots gathering problem
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
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Given a nonempty and finite multiset of points P in R^d, the Euclidean median of P, denoted M(P), is a point in R^d that minimizes the sum of the Euclidean (@?"2) distances from M(P) to the points of P. In two or more dimensions, the Euclidean median (otherwise known as the Weber point) is unstable; small perturbations at only a few points of P can result in an arbitrarily large relative change in the position of the Euclidean median. This instability motivates us to consider alternate notions for location functions that approximate the minimum sum of distances to the points of P while maintaining a fixed degree of stability. We introduce the projection median of a multiset of points in R^2 and compare it against the rectilinear (@?"1) median and the centre of mass, both in terms of approximation factor and stability. We show that a mobile facility located at the projection median of the positions of a set of mobile clients provides a good approximation of the mobile Euclidean median while ensuring both continuous motion and low relative velocity.